Mixing
relation between two positive definite matrices via similarity transformation.
$begingroup$
If $A$ and $B$ are two complex hermitian semi-definite matrices of order $Ntimes N$. Show that $exists$ an invertible matrix $Tinmathbb{C}^{Ntimes N}$ s.t
$T^H A T=Lambda_N$
$T^H B T= I_N$
where $Lambda_N$ is a diagonal matrix with positive entries and $I_N$ is the identity matrix.
linear-algebra
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
$endgroup$
add a comment |
$begingroup$
If $A$ and $B$ are two complex hermitian semi-definite matrices of order $Ntimes N$. Show that $exists$ an invertible matrix $Tinmathbb{C}^{Ntimes N}$ s.t
$T^H A T=Lambda_N$
$T^H B T= I_N$
where $Lambda_N$ is a diagonal matrix with positive entries and $I_N$ is the identity matrix.
linear-algebra
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
$endgroup$
add a comment |
$begingroup$
If $A$ and $B$ are two complex hermitian semi-definite matrices of order $Ntimes N$. Show that $exists$ an invertible matrix $Tinmathbb{C}^{Ntimes N}$ s.t
$T^H A T=Lambda_N$
$T^H B T= I_N$
where $Lambda_N$ is a diagonal matrix with positive entries and $I_N$ is the identity matrix.
linear-algebra
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
$endgroup$
If $A$ and $B$ are two complex hermitian semi-definite matrices of order $Ntimes N$. Show that $exists$ an invertible matrix $Tinmathbb{C}^{Ntimes N}$ s.t
$T^H A T=Lambda_N$
$T^H B T= I_N$
where $Lambda_N$ is a diagonal matrix with positive entries and $I_N$ is the identity matrix.
linear-algebra
linear-algebra
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
asked Jan 10 at 11:12
MD Afaque AzamMD Afaque Azam
62
62
add a comment |
add a comment |
1 Answer
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$begingroup$
Hint: Let $B=Q^{H}DQ$, be the eigen value decomposition of $B$ so $D$ is the diagonal then choose $T= Q^{H}D^{-1/2}$, and show that both relations hold.
answered Jan 10 at 13:38
zimbra314zimbra314
553212
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Hint: Let $B=Q^{H}DQ$, be the eigen value decomposition of $B$ so $D$ is the diagonal then choose $T= Q^{H}D^{-1/2}$, and show that both relations hold.
answered Jan 10 at 13:38
zimbra314zimbra314
553212
$endgroup$
add a comment |
$begingroup$
Hint: Let $B=Q^{H}DQ$, be the eigen value decomposition of $B$ so $D$ is the diagonal then choose $T= Q^{H}D^{-1/2}$, and show that both relations hold.
answered Jan 10 at 13:38
zimbra314zimbra314
553212
$endgroup$
add a comment |
$begingroup$
Hint: Let $B=Q^{H}DQ$, be the eigen value decomposition of $B$ so $D$ is the diagonal then choose $T= Q^{H}D^{-1/2}$, and show that both relations hold.
answered Jan 10 at 13:38
zimbra314zimbra314
553212
$endgroup$
Hint: Let $B=Q^{H}DQ$, be the eigen value decomposition of $B$ so $D$ is the diagonal then choose $T= Q^{H}D^{-1/2}$, and show that both relations hold.
answered Jan 10 at 13:38
zimbra314zimbra314
553212
answered Jan 10 at 13:38
zimbra314zimbra314
553212
answered Jan 10 at 13:38
zimbra314zimbra314
553212
answered Jan 10 at 13:38
zimbra314zimbra314
553212
553212
add a comment |
add a comment |
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